We see that the growth rates are tunable by parameter \(\beta \!\in \![0,1)\) . For example, by setting \(\beta \!=\!2/3\)  we obtain \(\mathcal {R}_T\!=\!\mathcal {O}(T^{7/12})\)  and \(\mathcal {V}_T\!=\!\mathcal {O}(T^{5/6})\) ; while with \(\beta \!=3/5\)  it is \(\mathcal {R}_T\!=\!\mathcal {O}(T^{3/5})\)  and \(\mathcal {V}_T\!=\!\mathcal {O}(T^{4/5})\) . These bounds improve the best-known results for the general constrained-OCO problem, while being comparable with results that consider special cases, such as knowing the (a priori fixed) time horizon \(T\)  [1]} or having only linearly-perturbed constraints [2]}.
